Turn off HDRI shadow casting
Sadly you cannot turn off shadows for emissive material or the world background.

You can control the shadows of any object in that objetc's shaders using the Light path Shadow ray input or by unchecking the Shadow option under Ray visibility in Object panel, but that reacts to all light sources.

The only solution is to use Light primitives, for which you can turn shadows on/off. If you create a big dense icosphere and you put a point-light in every vertex, you get a sky dome with even light distribution.

Using python you can access the HDRI texture pixels and drive every light's strength and color to match a point on the texture accordingly. How the pixel array works is explained here and figuring the math of turning a point on a dome into 2D point on texture is a matter of some trigonometry. You can use a script to be run once and set this for all the dome lights.

You can also add drivers to link and multiply the strength of those lights with some Scene property for better control. How to create drivers with python script is here.